On the box dimensions of graphs of typical continuous functions

Let X⊆R be a bounded set; we emphasize that we are not assuming that X is compact or Borel. We prove that for a typical (in the sense of Baire) uniformly continuous function f on X, the lower box dimension of the graph of f is as small as possible and the upper box dimension of the graph of f is as big as possible. We also prove a local version of this result. Namely, we prove that for a typical uniformly continuous function f on X, the lower local box dimension of the graph of f at all points x∈X is as small as possible and the upper local box dimension of the graph of f at all points x∈X is as big as possible.